Numerical Stability

Goal

Provide a repeatable checklist and script-driven checks to keep time-dependent simulations stable and defensible.

Requirements

• Python 3.8+
• NumPy (for matrix_condition.py and von_neumann_analyzer.py)
• See scripts/requirements.txt for dependencies

Inputs to Gather

Decision Guidance

Choosing Explicit vs Implicit

Is the problem stiff (fast + slow dynamics)?
├── YES → Use implicit or IMEX scheme
│         └── Check conditioning with matrix_condition.py
└── NO → Is CFL/Fourier satisfied with reasonable dt?
    ├── YES → Use explicit scheme (cheaper per step)
    └── NO → Consider implicit or reduce dx

Stability Limit Quick Reference

Multi-dimensional correction: For d dimensions, diffusion limit is Fo ≤ 1/(2d).

Script Outputs (JSON Fields)

Workflow

1. Identify dominant physics (advection vs diffusion vs reaction)
2. Run CFL checker with scripts/cfl_checker.py
3. Compare to limits and adjust dt if needed
4. Check stiffness with scripts/stiffness_detector.py if multiple scales
5. Analyze custom schemes with scripts/von_neumann_analyzer.py
6. Check conditioning with scripts/matrix_condition.py for implicit solves
7. Document the stability verdict and recommended time step

Conversational Workflow Example

User: My phase-field simulation is blowing up after 100 steps. I'm using explicit Euler with dx=0.01, dt=1e-4, and diffusivity D=1e-3.

Agent workflow:

1. Check stability criteria:
   bashCopy

   python3 scripts/cfl_checker.py --dx 0.01 --dt 1e-4 --diffusivity 1e-3 --dimensions 2 --json
   

2. Interpret results:
   • Fourier number: Fo = 1e-3 × 1e-4 / (0.01)² = 1.0
   • 2D limit: Fo ≤ 0.25
   • Violation: Fo = 1.0 > 0.25, unstable!
3. Recommend fix:
   • Reduce dt to 2.5e-5 (to get Fo = 0.25)
   • Or increase dx, or switch to implicit

Pre-Simulation Stability Checklist

• Identify dominant physics and nondimensional groups
• Compute CFL/Fourier/Reaction numbers with cfl_checker.py
• If explicit and limit violated, reduce dt or change scheme
• If stiffness ratio > 1000, select implicit/stiff integrator
• For custom schemes, verify amplification factor ≤ 1
• Document stability reasoning with inputs and outputs

CLI Examples

# Check CFL/Fourier for 2D diffusion-advection
python3 scripts/cfl_checker.py --dx 0.1 --dt 0.01 --velocity 1.0 --diffusivity 0.1 --dimensions 2 --json

# Von Neumann analysis for custom 3-point stencil
python3 scripts/von_neumann_analyzer.py --coeffs 0.2,0.6,0.2 --dx 1.0 --nk 128 --json

# Detect stiffness from eigenvalue estimates
python3 scripts/stiffness_detector.py --eigs=-1,-1000 --json

# Check matrix conditioning for implicit system
python3 scripts/matrix_condition.py --matrix A.npy --norm 2 --json

Error Handling

Interpretation Guidance

Limitations

• Explicit schemes only for CFL/Fourier checks (implicit is unconditionally stable)
• Von Neumann analysis assumes linear, constant-coefficient, periodic BCs
• Stiffness detection requires eigenvalue estimates from user

References

• references/stability_criteria.md - Decision thresholds and formulas
• references/common_pitfalls.md - Frequent failure modes and fixes
• references/scheme_catalog.md - Stability properties of common schemes

Version History

• v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
• v1.0.0: Initial release with 4 stability analysis scripts